Cohomology

Dual cochain constructions yield invariants and additional operations such as cup products.

Different ways to understand it

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Exact sequences
At each position, the incoming image equals the outgoing kernel.
Persistent homology
Homological features are tracked across a nested family of spaces or data scales.
Simplicial complexes
Simplices fitted together along faces provide combinatorial models of spaces.
Topological spaces
A topology specifies open sets closed under arbitrary unions and finite intersections.
Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.